C₃×#10345 #10628952

order	768
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10345

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃	#120
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₃
prime radical	C₂⁴×C₃	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂³×C₃×C₄ ◁ G

640 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3)(4,5,7,9)(6,10,11,8)(12,13)(14,16)(15,17)(18,19)(20,21,23,25)(22,28,29,35)(24,32,31,27)(26,33,34,30);(4,6,7,11)(5,8,9,10)(12,14)(13,16)(15,18)(17,19)(21,25)(26,34)(27,32)(28,35);(16,19)(20,22,23,29)(21,26,25,34)(24,33,31,30)(27,35,32,28)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3333);